A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244

Offset

1,1

Comments

First differs from A177425 in lacking 360.

First differs from A182854 in lacking 360.

These are the Heinz numbers of the partitions counted by A182473.

Links

Table of n, a(n) for n=1..57.

Example

The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}

Mathematica

Select[Range[100], PrimeNu[#]==2&&UnsameQ@@Last/@FactorInteger[#]&]

Crossrefs

The case of any multiplicities is A007774, counts A002133.

These partitions are counted by A182473.

The case of equal exponents is A367590, counts A367588.

A000041 counts integer partitions, strict A000009.

A091602 counts partitions by greatest multiplicity, least A243978.

A098859 counts partitions with distinct multiplicities, ranks A130091.

A116608 counts partitions by number of distinct parts.

Cf. A071625, A072233, A072774, A109297, A367580.

Sequence in context: A360248 A317616 A332785 * A177425 A182854 A348097

Adjacent sequences: A367586 A367587 A367588 * A367590 A367591 A367592

Keyword

nonn

Author

Gus Wiseman, Dec 01 2023

Status

approved